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Selecta Mathematica

, Volume 22, Issue 1, pp 1–25 | Cite as

Quantization of line bundles on lagrangian subvarieties

  • Vladimir Baranovsky
  • Victor Ginzburg
  • Dmitry Kaledin
  • Jeremy Pecharich
Article

Abstract

We apply the technique of formal geometry to give a necessary and sufficient condition for a line bundle supported on a smooth Lagrangian subvariety to deform to a sheaf of modules over a fixed deformation quantization of the structure sheaf of an algebraic symplectic variety.

Mathematics Subject Classification

53D55 14D21 

Notes

Acknowledgments

We are grateful to Dima Arinkin for helpful remarks and to Pierre Schapira for historical comments. The first author was supported by a Simons Foundation Collaboration Grant. The second author was supported in part by the NSF Grant DMS-1001677. The third author was partially supported by the RFBR Grant  12-01-33024, Russian Federation Government Grant, ag. 11.G34.31.0023, and the Dynasty Foundation Award. The fourth author would like to thank K. Behrend and B. Fantechi for many inspiring discussions on quantization. He is grateful to the Mathematical Science Research Institute, Berkeley, for hospitality during his stay in the Spring of 2013. The work on the paper was also partially supported by the NSF Grant DMS-0932078 000.

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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Vladimir Baranovsky
    • 1
  • Victor Ginzburg
    • 2
  • Dmitry Kaledin
    • 3
  • Jeremy Pecharich
    • 4
  1. 1.Department of MathematicsUniversity of California at IrvineIrvineUSA
  2. 2.Department of MathematicsUniversity of ChicagoChicagoUSA
  3. 3.Algebraic Geometry SectionSteklov Mathematical InstituteMoscowRussia
  4. 4.Department of MathematicsPomona CollegeClaremontUSA

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